Given a vector \(q \in \mathbb {R}^n\)  and an \(n \times n\)  matrix \(M\)  over the reals, the linear complementarity problem [1]} asks whether there exist \(w,z \in \mathbb {R}^n\)  satisfying \(w-M z = q\) , \(w,z\ge 0\) , and \(w.z=0\) , where \(x \ge 0\)  means that \(x\)  belongs to \(\mathbb {R}^n_+\) , the non-negative orthant (or cone) of \(\mathbb {R}^n\) , and \(w.z\)  is the scalar product of \(w\)  and \(z\) . Let LCP\((q,M)\)  denote the set of solutions for \(q\) . When LCP\((q,M)\)  is non-empty for every vector \(q\)  in \(\mathbb {R}^n\) , the matrix \(M\)  is called a Q-matrix. When LCP\((q,M)\)  is reduced to a singleton, that is there exists a unique solution for every \(q\) , \(M\)  is called a P-matrix.
While a satisfying characterization for P-matrices has been known since the late fifties [2]}, the same cannot be said for Q-matrices despite several insights and serious attempts over the past sixty years.
